1. Field of the Invention
The present invention relates generally to the field of signal processing networks, and more particularly to a method and apparatus for modeling a network capable of processing analog and/or digital signals, with linear or non-linear response characteristics, capable of receiving multichannel inputs and outputs that may o may not be history dependent.
2. Description of the Relevant Art
Modeling and synthesis techniques involve determination of the relationship or relationships between selected inputs to a network and the desired or actual output of the network in response to those inputs. The techniques are fundamental to the arts of Computer Aided Design (CAD), Computer Aided Engineering (CAE), Test System Design, Simulation, Etc.
As used herein, synthesis of a network involves deriving a mechanism (e.g., one or more functions, values stored in a look-up table, etc.) which represents the network's response to certain inputs or ranges of inputs. Similarly, modeling a network involves utilizing the synthesized mechanism in a system in place of the actual network.
Problems of modeling and synthesis of networks fall into several categories of complexity which limit the applicability of particular existing modeling and synthesis techniques. A network may operate on analog or digital signals, or both. The response of the network may be linear or non-linear. The present response may depend on one or more of the past inputs or responses of the network. The network may operate on one or more channels of input. Any and all of these network characteristics increase the complexity of the modeling and synthesis of the network and, consequently, reduce the number of available techniques for modeling and synthesis.
There exist many methods of modeling and synthesizing linear networks, from linear regression to complex numerical modeling. The number of methods available is significantly reduced when the network's response is non-linear. There are three major methods traditionally used to model the input/output relationships of the non-linear network. The first is the quasi-linear methods of the Taylor Series Perturbation and Describing functions. The second is the differential equation method. The third is the integral equation or functional (or Wiener) method. Each of these methods is briefly described below.
The Taylor Series Perturbation method involves establishing a dynamic operating point about which a Taylor Series is used to represent small signal deviations as perturbations, i.e., network response to an input. The disadvantages of the Taylor Series method involve the fact that the method depends upon the existence of derivatives of the response function (or "gain") to determine the functional response of the network. If such derivatives do not exist, the algorithm used fails to converge. This implies that only "smooth" non-linearities can be tolerated. An example of an application of this method is the SPICE computer program for use in circuit analysis.
The describing function method involves producing a linearized model for each contribution to the output of the network. Sinusoidal describing functions, for example, form a linear network that represent an output as an undistorted sinusoid of the same frequency as the input signal and with a "gain" corresponding to the gain of the actual network at the excitation frequency. The method requires a new and possibly different model for each sinusoidal amplitude and frequency.
A more general describing function can be constructed by using broadband excitation representative of the actual input signal and estimating the gain and phase shift for all pertinent frequencies by means of the input/output cross-spectral density, S.sub.vx (.omega.), and the input auto-spectral density, S.sub.xx (.omega.). This results in a linear transfer function estimate, H(.omega.), where ##EQU1## where E{.}is the expected value.
This yields a transfer function (gain and phase shift) at a given frequency based solely on the response, v(t), that is phase coherent with the excitation, x(t). The total output energy consists of the phase coherent signal and a residual output signal, n(t), that is uncorrelated with the excitation x(t). H(.omega.) represents the best (mean square error) linear estimate of a transfer function.
The differential equation method involves the direct solution of non-linear differential equations. Because this method is very computation intensive, its implementation is mainly restricted to high-speed computer applications. Further, its results provide a response solution, v(t), only for a specific excitation signal, x(t). Finally, this method may also suffer from convergence problems similar to those mentioned above, depending on the program used for differential equation evaluation.
The integral equation or functional method (developed by Norbert Wiener) involves consideration of not only the instantaneous non-linear distortions of present input values, but also non-linear interactions, over time, of past and present data. This is a characteristic of the general network problem involving memory (or history dependence).
The functional method represents memory with an infinite set of orthogonal filters such as a bank of filters having the different order Laguerre functions as their impulse responses. The complete set of filter outputs, {u.sub.i (t)}, at a given point in time constitute, in some sense, a complete history of the excitation signal, x(t). Specifically, ##EQU2## From which a general non-linear function may be constructed using the following multinomial power series ##EQU3## Substituting the expression for u.sub.i (t) yields v(t) in the form of a Volterra equation of the first kind ##EQU4## where ##EQU5## K.sub.n (.) is the nth order impulse response of the network, an n-dimensional impulse response function that has a corresponding n-dimensional Fourier transform EQU k.sub.n (T.sub.1,T.sub.2,...T.sub.n).rarw..fwdarw.K.sub.n (.omega..sub.1,.omega..sub.2,....omega..sub.n)
which represents the broadband or multifrequency response of the network to all combinations of frequencies taken n at a time. The resulting transfer function K.sub.n (.) gives the gain and the phases at each of the interacting frequencies.
Although the functional method is comprehensive, its applications have been limited to modeling mildly non-linear networks requiring the determination of a small set of low order terms, for example as operational amplifier networks with small third order distortion. In the case of severe or non-integer-power non-linearities, the power series expansion is grossly inefficient because of the large number of terms required.
To compound the complexities and inadequacies of the above-mentioned methods, the general network solution must not be limited by any distinctions between analog and digital signals, especially when the inputs may be in mixed analog and digital form. These networks, representing the interface between the digital computer domain and the "natural" analog world, are characterized by "hard" non-linearities such as thresholding elements in analog to digital converters. Also, the logic networks are grossly non-linear, so that the combination of both types in a single network present formidable modeling problems. None of the above-mentioned methods have been able to adequately deal with the mixed analog and digital signal cases.
Finally, a realistic model of a network must be able to represent the case of multichannel inputs. For example, when modeling the response of a semiconductor element, time-varying values such as current density, temperature, electric field, etc., all contribute to the response characteristics of the device. Proper consideration of multichannel inputs increases the number of possible bit interactions exponentially, and correspondingly increases the complexity of the synthesis and resulting model.
There is a present need in the art for a method and system for modeling and synthesis of networks of mixed analog and/or digital signals having linear or non-linear responses which may receive a plurality of input channels and whose inputs may be memory dependent, which overcome the limitations and disadvantages of the prior art. The present invention provides such a method and system, as further described in detail below.